A 4"x4"x4" cube is painted and then cut into sixty-four 1"x1"x1" cubes. A unit cube is then randomly selected and rolled. What is the probability that the top face of the rolled cube is painted? Express your answer as a common fraction.

Now, originally, I tohught the answer was 96/384, but I realized it was wrong since some sides (the corner cubes) have more than one painted side.

Dec 13th 2007, 04:02 PM

Plato

Quote:

Originally Posted by help1

A 4"x4"x4" cube is painted and then cut into sixty-four 1"x1"x1" cubes. A unit cube is then randomly selected and rolled. What is the probability that the top face of the rolled cube is painted? Express your answer as a common fraction.
Now, originally, I tohught the answer was 96/384, but I realized it was wrong since some sides (the corner cubes) have more than one painted side.

There are cubes with no painted sides.
There are cubes with one painted side.
There are cubes with two painted sides.
There are cubes with three painted sides.
How many of each?

Dec 13th 2007, 04:05 PM

galactus

This is actually a cool problem. There are patterns to the number of painted faces on a nXn cube.

For a 3X3X3 cube, there are 8 cubes with 3 painted faces. Any cube will always have 8 cubes with 3 painted faces. Those are the corners.

There are 12 cubes with 2 faces painted. 6 cubes with 1 face painted and 1 with no faces painted(the one in the dead center).